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A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws
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Year:    2014

Communications in Computational Physics, Vol. 16 (2014), Iss. 3 : pp. 718–763

Abstract

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique not only reduces the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.181113.140314a

Communications in Computational Physics, Vol. 16 (2014), Iss. 3 : pp. 718–763

Published online:    2014-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    46

Keywords:   

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    Journal of Scientific Computing, Vol. 83 (2020), Iss. 2

    https://doi.org/10.1007/s10915-020-01209-w [Citations: 3]

  74. Discontinuous Galerkin Methods for Compressible and Incompressible Flows on Space–Time Adaptive Meshes: Toward a Novel Family of Efficient Numerical Methods for Fluid Dynamics

    Fambri, Francesco

    Archives of Computational Methods in Engineering, Vol. 27 (2020), Iss. 1 P.199

    https://doi.org/10.1007/s11831-018-09308-6 [Citations: 13]

  75. A direct Arbitrary-Lagrangian–Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D

    Boscheri, Walter | Dumbser, Michael

    Journal of Computational Physics, Vol. 275 (2014), Iss. P.484

    https://doi.org/10.1016/j.jcp.2014.06.059 [Citations: 106]

  76. High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

    Gaburro, Elena | Boscheri, Walter | Chiocchetti, Simone | Klingenberg, Christian | Springel, Volker | Dumbser, Michael

    Journal of Computational Physics, Vol. 407 (2020), Iss. P.109167

    https://doi.org/10.1016/j.jcp.2019.109167 [Citations: 72]

  77. A MOOD-like compact high order finite volume scheme with adaptive mesh refinement

    Loubère, Raphaël | Turpault, Rodolphe | Bourriaud, Alexandre

    Applied Mathematics and Computation, Vol. 443 (2023), Iss. P.127792

    https://doi.org/10.1016/j.amc.2022.127792 [Citations: 1]

  78. Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

    Zanotti, Olindo | Fambri, Francesco | Dumbser, Michael | Hidalgo, Arturo

    Computers & Fluids, Vol. 118 (2015), Iss. P.204

    https://doi.org/10.1016/j.compfluid.2015.06.020 [Citations: 118]

  79. A relaxed a posteriori MOOD algorithm for multicomponent compressible flows using high-order finite-volume methods on unstructured meshes

    Tsoutsanis, Panagiotis | Pavan Kumar, Machavolu Sai Santosh | Farmakis, Pericles S.

    Applied Mathematics and Computation, Vol. 437 (2023), Iss. P.127544

    https://doi.org/10.1016/j.amc.2022.127544 [Citations: 2]

  80. A high-order relativistic two-fluid electrodynamic scheme with consistent reconstruction of electromagnetic fields and a multidimensional Riemann solver for electromagnetism

    Balsara, Dinshaw S. | Amano, Takanobu | Garain, Sudip | Kim, Jinho

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    https://doi.org/10.1016/j.jcp.2016.05.006 [Citations: 43]

  81. A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

    Abreu, Eduardo | Agudelo, Jorge | Pérez, John

    Journal of Computational and Applied Mathematics, Vol. 437 (2024), Iss. P.115465

    https://doi.org/10.1016/j.cam.2023.115465 [Citations: 0]

  82. Reprint of: Direct Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes

    Gaburro, Elena | Dumbser, Michael | Castro, Manuel J.

    Computers & Fluids, Vol. 169 (2018), Iss. P.263

    https://doi.org/10.1016/j.compfluid.2018.03.051 [Citations: 3]

  83. Space-Time Adaptive ADER-DG Finite Element Method with LST-DG Predictor and a posteriori Sub-cell WENO Finite-Volume Limiting for Simulation of Non-stationary Compressible Multicomponent Reactive Flows

    Popov, I. S.

    Journal of Scientific Computing, Vol. 95 (2023), Iss. 2

    https://doi.org/10.1007/s10915-023-02164-y [Citations: 4]

  84. Efficient implementation of space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows

    Popov, Ivan S

    Journal of Physics: Conference Series, Vol. 1740 (2021), Iss. 1 P.012059

    https://doi.org/10.1088/1742-6596/1740/1/012059 [Citations: 0]

  85. High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

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    Communications on Applied Mathematics and Computation, Vol. 6 (2024), Iss. 3 P.1954

    https://doi.org/10.1007/s42967-023-00355-w [Citations: 0]

  86. Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers

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    Journal of Computational Physics, Vol. 299 (2015), Iss. P.687

    https://doi.org/10.1016/j.jcp.2015.07.012 [Citations: 81]

  87. An almost fail-safe a-posteriori limited high-order CAT scheme

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    Journal of Computational Physics, Vol. 498 (2024), Iss. P.112650

    https://doi.org/10.1016/j.jcp.2023.112650 [Citations: 3]